question on factoring..?

Question

how do I factor x^4+2x^2+9 and x^4+4x^2+16

THANKS!!!!

to riya: i don't get it.. what was you basis on doing on what you just did?

Answer 1

x^4+2x^2+9
=(x^2)^2+6x^2+(3)^2-4x^2
=(x^2+3)^2-(2x)^2
=(x^2+3+2x)(x^2+3-2x)
=(x^2+2x+3)(x^2-2x+3)

x^4+4x^2+16
=(x^2)^2+2*x^2*4+(4)^2-4x^2
=(x^2+4)^2-(2x)^2
=(x^2+4+2x)(x^2+4-2x) [ identity a^2-b^2=
(a+b)(a-b) ]
=(x^2+2x+4)(x^2-2x+4) [re-arranging according to the power of x]

Answer 2

To factor these expressions, you need to use what it called "completing the square". To start, you need to set the constant to the side and just look at the first two factors.

x^4 + 2x^2 + ______ + 9 - ______

[make sure to leave 9 to the side, don't forget about it and drop it from the equation!]

We want to fill in that first blank with a number so that the first expression (excluding the 9) will give us a perfect square.
However, in order for the expression to remain the same, we must again subtract that number from our original constant.

In this case, the number we are looking for is simply 1. A 1 will give us a perfect square with the first two factors. By adding a 1, we get:

(x^4 + 2x ^2 + 1) +9 - 1
(x^2 + 1)^2 +8

We are simply manipulating the expression to make it possible to factor. We didn't actually add anything to the expression or change it. We added 1 to the first grouping of factors and subtracted 1 from the second.

To check the answer, multiply it out:
(x^2 +1) (x^2 + 1) + 8
x^4 + x^2 + x^2 + 1 + 8
x^4 + 2x^2 + 9

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The second one can be completed in the same manor:

(x^4 + 4x^2 + _____ ) + 16 - _____

In this case, the blank needs to be filled in with a number that will make the first set of brackets into a perfect square.

This example is a little harder because the number is not simply 1. We want the expression to look as follows:

(x^2 + 2)^2 + 16 - 4

Notice that inside the brackets, there is a "2", while a 4 is being subtracted from the 16. That's because when we multiply the perfect square out, we square that 2, making it "4" that we've actually added to the equation. Because of that, we must subtract that 4 from the equation.

The final answer is (x^2 + 2) ^2 + 12.

Answer 3

x^4 + 2x² + 9
= (x^4 + 2x² + 1) - 1 + 9
= (x² + 1) ² + 8

x^4 + 4x² + 16
= (x^4 + 4x² + 4) - 4 + 16
= (x² + 2) ² + 12

Answer 4

x^4+2x^2+9
={[(x^2)^2]+(2*1*x^2)+1}+(9-1)
=(x^2+1)^2+8

x^4+4x^2+16
={[(x^2)^2]+(2*2*x^2)+(2^2)}+(16-4)
=(x^2+2)^2+12